"What happened in Mississippi was appalling," said Cruz, the vice chair of the National Republican Senatorial Committee. "Primaries are always rough and tumble. But the conduct of the Washington, D.C., machine in the Mississippi runoff was incredibly disappointing."

After losing a plurality of votes in the initial Republican primary, Cochran defeated the Tea Party-aligned McDaniel by about 7,600 votes, according to official certified results finalized late Monday night. Cochran employed an unusual strategy on the path to victory, courting the votes of many traditional Democrats, including African-Americans.

Cruz said in the interview it was clear McDaniel had won a "sizable majority" of votes from Mississippi Republican voters, and he accused Cochran's allies of "racially charged" ads and conduct that tipped the balance in Cochran's favor. He said the McDaniel campaign's claims of voter fraud should be "vigorously investigated."

"But even more troubling, in the past week or so, we've seen serious allegations of voter fraud, and I very much hope that no Republican was involved in voter fraud," Cruz said. "But these allegations need to be vigorously investigated, and anyone involved in criminal conduct should be prosecuted. The voters of Mississippi deserve to know the truth."